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This question already has an answer here:

Where is my mistake ? (In the field of real numbers)

$$ (-1)^3=-1 \to \sqrt[3]{-1}=-1 $$

$$\sqrt[3]{-1}=-1=(-1)^{\frac{1}{3}}=(-1)^{\frac{2}{6}}=\sqrt[6]{(-1)^2}=\sqrt[6]{(1)^2}=1$$

$$-1=^?1 $$

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marked as duplicate by kingW3, Matthew Towers, Vidyanshu Mishra, Namaste, Joel Reyes Noche Dec 19 '16 at 12:52

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    $\begingroup$ Easier $-1=(-1)^{2/2}=\sqrt{(-1)^2}=1$. But also $(-1)^{2/2}=(\sqrt{-1})^2$... ooops. $\endgroup$ – A.Γ. Nov 26 '16 at 10:55
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    $\begingroup$ This question has been asked here approximately $200$ times. $\endgroup$ – barak manos Nov 26 '16 at 10:59
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    $\begingroup$ For the record, the law $a^{bc}=(a^b)^c$ is only valid when $a$ is real and positive and $b$ is real. See this answer You implicitly used this law without satisfying the condition $a>0$ as $(-1)^{\frac26}=((-1)^2)^{\frac16}$, which is wrong. $\endgroup$ – Marc van Leeuwen Nov 26 '16 at 12:07
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That is the same error as in $1=\sqrt{1}=\sqrt{(-1)^2}=-1$

That is based on the fact: $1^\frac{1}{2} = \sqrt{1} = \pm 1$
The same is true for $1^\frac{1}{6} = \sqrt[6]{1} = \pm 1$
So you get $-1=\pm 1$ and that is true.

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The mistake lies in assuming that $(1)^{\frac{1}{6}}=1$, this is because, $1$ has two sixth roots in the field of real numbers, $1,-1$

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  • $\begingroup$ No, $1$ being real and positive, $1^{\frac16}=1$ is perfectly well defined and unique. Nobody ever construes $e^{1/3}$ to be a (possibly!) non-real number or $e^{1/2}$ to be possibly negative. $\endgroup$ – Marc van Leeuwen Nov 26 '16 at 12:13
  • $\begingroup$ @MarcvanLeeuwen But, arent there 2 sixth roots of unity in the field of reals? $\endgroup$ – vidyarthi Nov 27 '16 at 17:53
  • $\begingroup$ Yes, but the set of $n$-th roots of unity is not produced by raising $1$ to the power $1/n$, which just gives $1$. Some people will write the imaginary unit $\mathbf i$ as $\sqrt{-1}$ (though most disapprove of this notation) even fewer would write $\mathbf i=(-1)^{\frac12}$, but hardly anybody would write $\mathbf i=1^{\frac14}$ or $\mathbf i\in1^{\frac14}$. $\endgroup$ – Marc van Leeuwen Nov 27 '16 at 19:02
  • $\begingroup$ @MarcvanLeeuwen So, the mistake lies in notation, not in the reasoning, right? $\endgroup$ – vidyarthi Nov 27 '16 at 19:04
  • $\begingroup$ Your mistake lies in interpreting exponential notation as being set-valued, while it is single valued. However the mistake of OP is either to introduce a non-integer power of $-1$ in the first place (which I would consider not defined at all), or for those who would admit this with some choice of definition, to rewrite $a^{p/q}$ as $\sqrt[q]{a^p}$ (with $a=-1$, $p=2$, $q=6$), which is definitely not valid when $a<0$. $\endgroup$ – Marc van Leeuwen Nov 27 '16 at 19:12

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